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Unit 5 – Systems of Linear Equations and Inequalities

This unit begins by ensuring that students understand that solutions to equations are points that make the equation true, while solutions to systems make all equations (or inequalities) true.  Graphical and substitution methods for solving systems are reviewed before the development of the Elimination Method.  Modeling with systems of equations and inequalities is stressed.  Finally, we develop the idea of using graphs to help solve equations.

Solutions to Systems and Solving by Graphing

LESSON/HOMEWORK

LECCIÓN/TAREA

LESSON VIDEO

EDITABLE LESSON

EDITABLE KEY

Solving Systems by Substitution

Properties of Systems and Their Solutions

The Method of Elimination

Modeling with Systems of Equations

Solving Equations Graphically

Solving Systems of Inequalities

Modeling with Systems of Inequalities

Unit Review

Unit #5 Review – Systems of Linear Equations and Inequalities

UNIT REVIEW

REPASO DE LA UNIDAD

EDITABLE REVIEW

Unit #5 Assessment Form A

EDITABLE ASSESSMENT

Unit #5 Assessment Form B

Unit #5 Assessment Form C

Unit #5 Assessment.Form D

Unit #5 Exit Tickets

Unit #5 Mid-Unit Quiz (Through Lesson #4) – Form A

Unit #5 Mid-Unit Quiz (Through Lesson #4) – Form B

Unit #5 Mid-Unit Quiz (Through Lesson #4) – Form C

U05.AO.01 – Solving Equations Graphically – Extra Practice (After Lesson #6)

EDITABLE RESOURCE

U05.AO.02 – Additional Modeling with Linear Systems

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5.1 Solve Systems of Equations by Graphing

Learning objectives.

By the end of this section, you will be able to:

• Determine whether an ordered pair is a solution of a system of equations
• Solve a system of linear equations by graphing
• Determine the number of solutions of linear system
• Solve applications of systems of equations by graphing

Be Prepared 5.1

Before you get started, take this readiness quiz.

• For the equation y = 2 3 x − 4 y = 2 3 x − 4 ⓐ is ( 6 , 0 ) ( 6 , 0 ) a solution? ⓑ is ( −3 , −2 ) ( −3 , −2 ) a solution? If you missed this problem, review Example 2.1 .
• Find the slope and y -intercept of the line 3 x − y = 12 3 x − y = 12 . If you missed this problem, review Example 4.42 .
• Find the x - and y -intercepts of the line 2 x − 3 y = 12 2 x − 3 y = 12 . If you missed this problem, review Example 4.21 .

Determine Whether an Ordered Pair is a Solution of a System of Equations

In Solving Linear Equations and Inequalities we learned how to solve linear equations with one variable. Remember that the solution of an equation is a value of the variable that makes a true statement when substituted into the equation.

Now we will work with systems of linear equations , two or more linear equations grouped together.

System of Linear Equations

When two or more linear equations are grouped together, they form a system of linear equations.

We will focus our work here on systems of two linear equations in two unknowns. Later, you may solve larger systems of equations.

An example of a system of two linear equations is shown below. We use a brace to show the two equations are grouped together to form a system of equations.

A linear equation in two variables, like 2 x + y = 7, has an infinite number of solutions. Its graph is a line. Remember, every point on the line is a solution to the equation and every solution to the equation is a point on the line.

To solve a system of two linear equations, we want to find the values of the variables that are solutions to both equations. In other words, we are looking for the ordered pairs ( x , y ) that make both equations true. These are called the solutions to a system of equations .

Solutions of a System of Equations

Solutions of a system of equations are the values of the variables that make all the equations true. A solution of a system of two linear equations is represented by an ordered pair ( x , y ).

To determine if an ordered pair is a solution to a system of two equations, we substitute the values of the variables into each equation. If the ordered pair makes both equations true, it is a solution to the system.

Let’s consider the system below:

Is the ordered pair ( 2 , −1 ) ( 2 , −1 ) a solution?

The ordered pair (2, −1) made both equations true. Therefore (2, −1) is a solution to this system.

Let’s try another ordered pair. Is the ordered pair (3, 2) a solution?

The ordered pair (3, 2) made one equation true, but it made the other equation false. Since it is not a solution to both equations, it is not a solution to this system.

Example 5.1

Determine whether the ordered pair is a solution to the system: { x − y = −1 2 x − y = −5 { x − y = −1 2 x − y = −5

ⓐ ( −2 , −1 ) ( −2 , −1 ) ⓑ ( −4 , −3 ) ( −4 , −3 )

Determine whether the ordered pair is a solution to the system: { 3 x + y = 0 x + 2 y = −5 . { 3 x + y = 0 x + 2 y = −5 .

ⓐ ( 1 , −3 ) ( 1 , −3 ) ⓑ ( 0 , 0 ) ( 0 , 0 )

Determine whether the ordered pair is a solution to the system: { x − 3 y = −8 −3 x − y = 4 . { x − 3 y = −8 −3 x − y = 4 .

ⓐ ( 2 , −2 ) ( 2 , −2 ) ⓑ ( −2 , 2 ) ( −2 , 2 )

Solve a System of Linear Equations by Graphing

In this chapter we will use three methods to solve a system of linear equations. The first method we’ll use is graphing.

The graph of a linear equation is a line. Each point on the line is a solution to the equation. For a system of two equations, we will graph two lines. Then we can see all the points that are solutions to each equation. And, by finding what the lines have in common, we’ll find the solution to the system.

Most linear equations in one variable have one solution, but we saw that some equations, called contradictions, have no solutions and for other equations, called identities, all numbers are solutions.

Similarly, when we solve a system of two linear equations represented by a graph of two lines in the same plane, there are three possible cases, as shown in Figure 5.2 :

For the first example of solving a system of linear equations in this section and in the next two sections, we will solve the same system of two linear equations. But we’ll use a different method in each section. After seeing the third method, you’ll decide which method was the most convenient way to solve this system.

Example 5.2

How to solve a system of linear equations by graphing.

Solve the system by graphing: { 2 x + y = 7 x − 2 y = 6 . { 2 x + y = 7 x − 2 y = 6 .

Solve each system by graphing: { x − 3 y = −3 x + y = 5 . { x − 3 y = −3 x + y = 5 .

Solve each system by graphing: { − x + y = 1 3 x + 2 y = 12 . { − x + y = 1 3 x + 2 y = 12 .

The steps to use to solve a system of linear equations by graphing are shown below.

To solve a system of linear equations by graphing.

• Step 1. Graph the first equation.
• Step 2. Graph the second equation on the same rectangular coordinate system.
• Step 3. Determine whether the lines intersect, are parallel, or are the same line.
• If the lines intersect, identify the point of intersection. Check to make sure it is a solution to both equations. This is the solution to the system.
• If the lines are parallel, the system has no solution.
• If the lines are the same, the system has an infinite number of solutions.

Example 5.3

Solve the system by graphing: { y = 2 x + 1 y = 4 x − 1 . { y = 2 x + 1 y = 4 x − 1 .

Both of the equations in this system are in slope-intercept form, so we will use their slopes and y -intercepts to graph them. { y = 2 x + 1 y = 4 x − 1 { y = 2 x + 1 y = 4 x − 1

Solve each system by graphing: { y = 2 x + 2 y = − x − 4 . { y = 2 x + 2 y = − x − 4 .

Solve each system by graphing: { y = 3 x + 3 y = − x + 7 . { y = 3 x + 3 y = − x + 7 .

Both equations in Example 5.3 were given in slope–intercept form. This made it easy for us to quickly graph the lines. In the next example, we’ll first re-write the equations into slope–intercept form.

Example 5.4

Solve the system by graphing: { 3 x + y = −1 2 x + y = 0 . { 3 x + y = −1 2 x + y = 0 .

We’ll solve both of these equations for y y so that we can easily graph them using their slopes and y -intercepts. { 3 x + y = −1 2 x + y = 0 { 3 x + y = −1 2 x + y = 0

Solve each system by graphing: { − x + y = 1 2 x + y = 10 . { − x + y = 1 2 x + y = 10 .

Solve each system by graphing: { 2 x + y = 6 x + y = 1 . { 2 x + y = 6 x + y = 1 .

Usually when equations are given in standard form, the most convenient way to graph them is by using the intercepts. We’ll do this in Example 5.5 .

Example 5.5

Solve the system by graphing: { x + y = 2 x − y = 4 . { x + y = 2 x − y = 4 .

We will find the x - and y -intercepts of both equations and use them to graph the lines.

Solve each system by graphing: { x + y = 6 x − y = 2 . { x + y = 6 x − y = 2 .

Try It 5.10

Solve each system by graphing: { x + y = 2 x − y = −8 . { x + y = 2 x − y = −8 .

Do you remember how to graph a linear equation with just one variable? It will be either a vertical or a horizontal line.

Example 5.6

Solve the system by graphing: { y = 6 2 x + 3 y = 12 . { y = 6 2 x + 3 y = 12 .

Try It 5.11

Solve each system by graphing: { y = −1 x + 3 y = 6 . { y = −1 x + 3 y = 6 .

Try It 5.12

Solve each system by graphing: { x = 4 3 x − 2 y = 24 . { x = 4 3 x − 2 y = 24 .

In all the systems of linear equations so far, the lines intersected and the solution was one point. In the next two examples, we’ll look at a system of equations that has no solution and at a system of equations that has an infinite number of solutions.

Example 5.7

Solve the system by graphing: { y = 1 2 x − 3 x − 2 y = 4 . { y = 1 2 x − 3 x − 2 y = 4 .

Try It 5.13

Solve each system by graphing: { y = − 1 4 x + 2 x + 4 y = − 8 . { y = − 1 4 x + 2 x + 4 y = − 8 .

Try It 5.14

Solve each system by graphing: { y = 3 x − 1 6 x − 2 y = 6 . { y = 3 x − 1 6 x − 2 y = 6 .

Example 5.8

Solve the system by graphing: { y = 2 x − 3 −6 x + 3 y = − 9 . { y = 2 x − 3 −6 x + 3 y = − 9 .

Try It 5.15

Solve each system by graphing: { y = − 3 x − 6 6 x + 2 y = − 12 . { y = − 3 x − 6 6 x + 2 y = − 12 .

Try It 5.16

Solve each system by graphing: { y = 1 2 x − 4 2 x − 4 y = 16 . { y = 1 2 x − 4 2 x − 4 y = 16 .

If you write the second equation in Example 5.8 in slope-intercept form, you may recognize that the equations have the same slope and same y -intercept.

When we graphed the second line in the last example, we drew it right over the first line. We say the two lines are coincident. Coincident lines have the same slope and same y -intercept.

Coincident Lines

Coincident lines have the same slope and same y -intercept.

Determine the Number of Solutions of a Linear System

There will be times when we will want to know how many solutions there will be to a system of linear equations, but we might not actually have to find the solution. It will be helpful to determine this without graphing.

We have seen that two lines in the same plane must either intersect or are parallel. The systems of equations in Example 5.2 through Example 5.6 all had two intersecting lines. Each system had one solution.

A system with parallel lines, like Example 5.7 , has no solution. What happened in Example 5.8 ? The equations have coincident lines , and so the system had infinitely many solutions.

We’ll organize these results in Figure 5.3 below:

Parallel lines have the same slope but different y -intercepts. So, if we write both equations in a system of linear equations in slope–intercept form, we can see how many solutions there will be without graphing! Look at the system we solved in Example 5.7 .

The two lines have the same slope but different y -intercepts. They are parallel lines.

Figure 5.4 shows how to determine the number of solutions of a linear system by looking at the slopes and intercepts.

Let’s take one more look at our equations in Example 5.7 that gave us parallel lines.

When both lines were in slope-intercept form we had:

Do you recognize that it is impossible to have a single ordered pair ( x , y ) ( x , y ) that is a solution to both of those equations?

We call a system of equations like this an inconsistent system . It has no solution.

A system of equations that has at least one solution is called a consistent system .

Consistent and Inconsistent Systems

A consistent system of equations is a system of equations with at least one solution.

An inconsistent system of equations is a system of equations with no solution.

We also categorize the equations in a system of equations by calling the equations independent or dependent . If two equations are independent equations , they each have their own set of solutions. Intersecting lines and parallel lines are independent.

If two equations are dependent, all the solutions of one equation are also solutions of the other equation. When we graph two dependent equations , we get coincident lines.

Independent and Dependent Equations

Two equations are independent if they have different solutions.

Two equations are dependent if all the solutions of one equation are also solutions of the other equation.

Let’s sum this up by looking at the graphs of the three types of systems. See Figure 5.5 and Figure 5.6 .

Example 5.9

Without graphing, determine the number of solutions and then classify the system of equations: { y = 3 x − 1 6 x − 2 y = 12 . { y = 3 x − 1 6 x − 2 y = 12 .

A system of equations whose graphs are parallel lines has no solution and is inconsistent and independent.

Try It 5.17

Without graphing, determine the number of solutions and then classify the system of equations.

{ y = − 2 x − 4 4 x + 2 y = 9 { y = − 2 x − 4 4 x + 2 y = 9

Try It 5.18

{ y = 1 3 x − 5 x − 3 y = 6 { y = 1 3 x − 5 x − 3 y = 6

Example 5.10

Without graphing, determine the number of solutions and then classify the system of equations: { 2 x + y = − 3 x − 5 y = 5 . { 2 x + y = − 3 x − 5 y = 5 .

A system of equations whose graphs are intersect has 1 solution and is consistent and independent.

Try It 5.19

{ 3 x + 2 y = 2 2 x + y = 1 { 3 x + 2 y = 2 2 x + y = 1

Try It 5.20

{ x + 4 y = 12 − x + y = 3 { x + 4 y = 12 − x + y = 3

Example 5.11

Without graphing, determine the number of solutions and then classify the system of equations. { 3 x − 2 y = 4 y = 3 2 x − 2 { 3 x − 2 y = 4 y = 3 2 x − 2

A system of equations whose graphs are coincident lines has infinitely many solutions and is consistent and dependent.

Try It 5.21

{ 4 x − 5 y = 20 y = 4 5 x − 4 { 4 x − 5 y = 20 y = 4 5 x − 4

Try It 5.22

{ −2 x − 4 y = 8 y = − 1 2 x − 2 { −2 x − 4 y = 8 y = − 1 2 x − 2

Solve Applications of Systems of Equations by Graphing

We will use the same problem solving strategy we used in Math Models to set up and solve applications of systems of linear equations. We’ll modify the strategy slightly here to make it appropriate for systems of equations.

Use a problem solving strategy for systems of linear equations.

• Step 1. Read the problem. Make sure all the words and ideas are understood.

Step 2. Identify what we are looking for.

Step 3. Name what we are looking for. Choose variables to represent those quantities.

Step 4. Translate into a system of equations.

Step 5. Solve the system of equations using good algebra techniques.

Step 6. Check the answer in the problem and make sure it makes sense.

Step 7. Answer the question with a complete sentence.

Step 5 is where we will use the method introduced in this section. We will graph the equations and find the solution.

Example 5.12

Sondra is making 10 quarts of punch from fruit juice and club soda. The number of quarts of fruit juice is 4 times the number of quarts of club soda. How many quarts of fruit juice and how many quarts of club soda does Sondra need?

Step 1. Read the problem.

We are looking for the number of quarts of fruit juice and the number of quarts of club soda that Sondra will need.

  Let f = f = number of quarts of fruit juice.      c = c = number of quarts of club soda

We now have the system. { f + c = 10 f = 4 c { f + c = 10 f = 4 c

The point of intersection (2, 8) is the solution. This means Sondra needs 2 quarts of club soda and 8 quarts of fruit juice.

Does this make sense in the problem?

Yes, the number of quarts of fruit juice, 8 is 4 times the number of quarts of club soda, 2.

Yes, 10 quarts of punch is 8 quarts of fruit juice plus 2 quarts of club soda.

Sondra needs 8 quarts of fruit juice and 2 quarts of soda.

Try It 5.23

Manny is making 12 quarts of orange juice from concentrate and water. The number of quarts of water is 3 times the number of quarts of concentrate. How many quarts of concentrate and how many quarts of water does Manny need?

Try It 5.24

Alisha is making an 18 ounce coffee beverage that is made from brewed coffee and milk. The number of ounces of brewed coffee is 5 times greater than the number of ounces of milk. How many ounces of coffee and how many ounces of milk does Alisha need?

Access these online resources for additional instruction and practice with solving systems of equations by graphing.

• Instructional Video Solving Linear Systems by Graphing
• Instructional Video Solve by Graphing

Section 5.1 Exercises

Practice makes perfect.

Determine Whether an Ordered Pair is a Solution of a System of Equations . In the following exercises, determine if the following points are solutions to the given system of equations.

{ 2 x − 6 y = 0 3 x − 4 y = 5 { 2 x − 6 y = 0 3 x − 4 y = 5

ⓐ ( 3 , 1 ) ( 3 , 1 ) ⓑ ( −3 , 4 ) ( −3 , 4 )

{ 7 x − 4 y = −1 −3 x − 2 y = 1 { 7 x − 4 y = −1 −3 x − 2 y = 1

ⓐ   ⓑ ( 1 , −2 ) ( 1 , −2 )

{ 2 x + y = 5 x + y = 1 { 2 x + y = 5 x + y = 1

ⓐ ( 4 , −3 ) ( 4 , −3 ) ⓑ ( 2 , 0 ) ( 2 , 0 )

{ −3 x + y = 8 − x + 2 y = −9 { −3 x + y = 8 − x + 2 y = −9

ⓐ ( −5 , −7 ) ( −5 , −7 ) ⓑ ( −5 , 7 ) ( −5 , 7 )

{ x + y = 2 y = 3 4 x { x + y = 2 y = 3 4 x

ⓐ ( 8 7 , 6 7 ) ( 8 7 , 6 7 ) ⓑ ( 1 , 3 4 ) ( 1 , 3 4 )

{ x + y = 1 y = 2 5 x { x + y = 1 y = 2 5 x

ⓐ ( 5 7 , 2 7 ) ( 5 7 , 2 7 ) ⓑ ( 5 , 2 ) ( 5 , 2 )

{ x + 5 y = 10 y = 3 5 x + 1 { x + 5 y = 10 y = 3 5 x + 1

ⓐ ( −10 , 4 ) ( −10 , 4 ) ⓑ ( 5 4 , 7 4 ) ( 5 4 , 7 4 )

{ x + 3 y = 9 y = 2 3 x − 2 { x + 3 y = 9 y = 2 3 x − 2

ⓐ ( −6 , 5 ) ( −6 , 5 ) ⓑ ( 5 , 4 3 ) ( 5 , 4 3 )

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.

{ 3 x + y = −3 2 x + 3 y = 5 { 3 x + y = −3 2 x + 3 y = 5

{ − x + y = 2 2 x + y = −4 { − x + y = 2 2 x + y = −4

{ −3 x + y = −1 2 x + y = 4 { −3 x + y = −1 2 x + y = 4

{ −2 x + 3 y = −3 x + y = 4 { −2 x + 3 y = −3 x + y = 4

{ y = x + 2 y = −2 x + 2 { y = x + 2 y = −2 x + 2

{ y = x − 2 y = −3 x + 2 { y = x − 2 y = −3 x + 2

{ y = 3 2 x + 1 y = − 1 2 x + 5 { y = 3 2 x + 1 y = − 1 2 x + 5

{ y = 2 3 x − 2 y = − 1 3 x − 5 { y = 2 3 x − 2 y = − 1 3 x − 5

{ − x + y = −3 4 x + 4 y = 4 { − x + y = −3 4 x + 4 y = 4

{ x − y = 3 2 x − y = 4 { x − y = 3 2 x − y = 4

{ −3 x + y = −2 4 x − 2 y = 6 { −3 x + y = −2 4 x − 2 y = 6

{ x + y = 5 2 x − y = 4 { x + y = 5 2 x − y = 4

{ x − y = 2 2 x − y = 6 { x − y = 2 2 x − y = 6

{ x + y = 2 x − y = 0 { x + y = 2 x − y = 0

{ x + y = 6 x − y = −8 { x + y = 6 x − y = −8

{ x + y = −5 x − y = 3 { x + y = −5 x − y = 3

{ x + y = 4 x − y = 0 { x + y = 4 x − y = 0

{ x + y = −4 − x + 2 y = −2 { x + y = −4 − x + 2 y = −2

{ − x + 3 y = 3 x + 3 y = 3 { − x + 3 y = 3 x + 3 y = 3

{ −2 x + 3 y = 3 x + 3 y = 12 { −2 x + 3 y = 3 x + 3 y = 12

{ 2 x − y = 4 2 x + 3 y = 12 { 2 x − y = 4 2 x + 3 y = 12

{ 2 x + 3 y = 6 y = −2 { 2 x + 3 y = 6 y = −2

{ −2 x + y = 2 y = 4 { −2 x + y = 2 y = 4

{ x − 3 y = −3 y = 2 { x − 3 y = −3 y = 2

{ 2 x − 2 y = 8 y = −3 { 2 x − 2 y = 8 y = −3

{ 2 x − y = −1 x = 1 { 2 x − y = −1 x = 1

{ x + 2 y = 2 x = −2 { x + 2 y = 2 x = −2

{ x − 3 y = −6 x = −3 { x − 3 y = −6 x = −3

{ x + y = 4 x = 1 { x + y = 4 x = 1

{ 4 x − 3 y = 8 8 x − 6 y = 14 { 4 x − 3 y = 8 8 x − 6 y = 14

{ x + 3 y = 4 −2 x − 6 y = 3 { x + 3 y = 4 −2 x − 6 y = 3

{ −2 x + 4 y = 4 y = 1 2 x { −2 x + 4 y = 4 y = 1 2 x

{ 3 x + 5 y = 10 y = − 3 5 x + 1 { 3 x + 5 y = 10 y = − 3 5 x + 1

{ x = −3 y + 4 2 x + 6 y = 8 { x = −3 y + 4 2 x + 6 y = 8

{ 4 x = 3 y + 7 8 x − 6 y = 14 { 4 x = 3 y + 7 8 x − 6 y = 14

{ 2 x + y = 6 −8 x − 4 y = −24 { 2 x + y = 6 −8 x − 4 y = −24

{ 5 x + 2 y = 7 −10 x − 4 y = −14 { 5 x + 2 y = 7 −10 x − 4 y = −14

{ x + 3 y = −6 4 y = − 4 3 x − 8 { x + 3 y = −6 4 y = − 4 3 x − 8

{ − x + 2 y = −6 y = − 1 2 x − 1 { − x + 2 y = −6 y = − 1 2 x − 1

{ −3 x + 2 y = −2 y = − x + 4 { −3 x + 2 y = −2 y = − x + 4

{ − x + 2 y = −2 y = − x − 1 { − x + 2 y = −2 y = − x − 1

Determine the Number of Solutions of a Linear System Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations.

{ y = 2 3 x + 1 −2 x + 3 y = 5 { y = 2 3 x + 1 −2 x + 3 y = 5

{ y = 1 3 x + 2 x − 3 y = 9 { y = 1 3 x + 2 x − 3 y = 9

{ y = −2 x + 1 4 x + 2 y = 8 { y = −2 x + 1 4 x + 2 y = 8

{ y = 3 x + 4 9 x − 3 y = 18 { y = 3 x + 4 9 x − 3 y = 18

{ y = 2 3 x + 1 2 x − 3 y = 7 { y = 2 3 x + 1 2 x − 3 y = 7

{ 3 x + 4 y = 12 y = −3 x − 1 { 3 x + 4 y = 12 y = −3 x − 1

{ 4 x + 2 y = 10 4 x − 2 y = −6 { 4 x + 2 y = 10 4 x − 2 y = −6

{ 5 x + 3 y = 4 2 x − 3 y = 5 { 5 x + 3 y = 4 2 x − 3 y = 5

{ y = − 1 2 x + 5 x + 2 y = 10 { y = − 1 2 x + 5 x + 2 y = 10

{ y = x + 1 − x + y = 1 { y = x + 1 − x + y = 1

{ y = 2 x + 3 2 x − y = −3 { y = 2 x + 3 2 x − y = −3

{ 5 x − 2 y = 10 y = 5 2 x − 5 { 5 x − 2 y = 10 y = 5 2 x − 5

Solve Applications of Systems of Equations by Graphing In the following exercises, solve.

Molly is making strawberry infused water. For each ounce of strawberry juice, she uses three times as many ounces of water. How many ounces of strawberry juice and how many ounces of water does she need to make 64 ounces of strawberry infused water?

Jamal is making a snack mix that contains only pretzels and nuts. For every ounce of nuts, he will use 2 ounces of pretzels. How many ounces of pretzels and how many ounces of nuts does he need to make 45 ounces of snack mix?

Enrique is making a party mix that contains raisins and nuts. For each ounce of nuts, he uses twice the amount of raisins. How many ounces of nuts and how many ounces of raisins does he need to make 24 ounces of party mix?

Owen is making lemonade from concentrate. The number of quarts of water he needs is 4 times the number of quarts of concentrate. How many quarts of water and how many quarts of concentrate does Owen need to make 100 quarts of lemonade?

Everyday Math

Leo is planning his spring flower garden. He wants to plant tulip and daffodil bulbs. He will plant 6 times as many daffodil bulbs as tulip bulbs. If he wants to plant 350 bulbs, how many tulip bulbs and how many daffodil bulbs should he plant?

A marketing company surveys 1,200 people. They surveyed twice as many females as males. How many males and females did they survey?

Writing Exercises

In a system of linear equations, the two equations have the same slope. Describe the possible solutions to the system.

In a system of linear equations, the two equations have the same intercepts. Describe the possible solutions to the system.

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

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Writing and Graphing Systems of Linear Equations

Lesson Narrative

This is the first of a series of lessons in which students review what they learned about systems of equations in middle school and develop new techniques for solving them.

In this lesson, students recall that a system of equations in two variables is a set of equations that represent multiple constraints in the same situation, and that a solution to the system is any pair of values that satisfy all the constraints simultaneously. Students also revisit the idea that the solution, if there is one, can be represented graphically as the intersection of the graphs of the equations.

Students write a system of equations to represent the quantities and constraints in each of several situations, find a solution that meets multiple constraints by graphing, and then interpret the solution in context. In the process, they reason both abstractly and quantitatively (MP2). As they analyze relationships mathematically and reflect on the results, students also engage in aspects of modeling (MP4).

Some students who recall the work on systems of equations from grade 8 may choose to solve the systems algebraically. This is appropriate and welcome, but it is not necessary to introduce the idea to the class here. Students will use algebra to solve systems starting in the next lesson.

Learning Goals

Teacher Facing

• Solve systems of linear equations by reasoning with tables and by graphing, and explain (orally and in writing) the solution method.
• Understand that the solution to a system of equations in two variables is a pair of values that simultaneously make both equations true, and that it is represented by the intersection point of the graphs of the equations.
• Understand that two (or more) equations that represent the constraints on the same quantities in the same situation form a system.

Student Facing

• Let’s recall what it means to solve a system of linear equations and how to do it by graphing.

Required Materials

• Graphing technology

Required Preparation

Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)

Learning Targets

• I can explain what we mean by “the solution to a system of linear equations” and can explain how the solution is represented graphically.
• I can explain what we mean when we refer to two equations as a system of equations.
• I can use tables and graphs to solve systems of equations.

CCSS Standards

Building On

• HSA-CED.A.3
• HSA-REI.C.6

Building Towards

Glossary Entries

A coordinate pair that makes both equations in the system true.

On the graph shown of the equations in a system, the solution is the point where the graphs intersect.

Two or more equations that represent the constraints in the same situation form a system of equations. 

Print Formatted Materials

For access, consult one of our IM Certified Partners .

Additional Resources

List of Lessons 1.1 Return to Algebra 1.2 Linear Inequalities 1.3 Absolute Value 1.4 Rewriting Equations Unit 1 Review Unit 1 Algebra Skillz Review 2.1 Function Notation 2.2 Functions/Relations 2.3 Slope and Rate of Change 2.4 Graphing Lines 2.5 Write Equations of Lines Unit 2 Review Unit 2 Algebra Skillz and SAT Review 3.1 Absolute Value Inequality 3.2 Absolute Value Graphs 3.3 Piecewise Functions Unit 3 Review Unit 3 Algebra Skillz 4.1 Solving Systems by Graphing 4.2 Solving Systems Algebraically 4.3 Systems of Inequalities Unit 4 Review 5.1 Graph in Vertex Form 5.2 Graph in Standard Form 5.3 Solve by Factoring 5.4 GCF and DoS 5.5 Solving by Square Roots Unit 5 Review Unit 5 Algebra Skillz and SAT Review 6.1 Imaginary and Complex Numbers 6.2 Operations on Complex Numbers 6.3 Completing the Square 6.4 Quadratic Formula Unit 6 Review Unit 6 Algebra Skillz SEMESTER EXAM 7.1 Properties of Exponents 7.2 Polynomial Division 7.3 Solving Polynomial Functions by Factoring 7.4 Graphs of Polynomial Functions 8.1 Evaluate Nth Roots 8.2 Properties of Rational Exponents 8.3 Function Operations and Composition 8.4 Inverse Operations 8.5 Graph Square and Cube Root Functions 8.6 Solving Radical Equations Unit 8 Review Unit 8 Algebra Skillz and SAT Review 9.1 Exponential Growth 9.2 Exponential Decay 9.3 The Number e 9.4 Intro to Logarithms 9.5 Properties of Logarithms 9.6 Solve Exponential and Log Equations Unit 9 Review Unit 9 Algebra Skillz 10.1 Graph Rational Functions 10.2 Multiply and Divide Rational Expressions 10.3 Add and Subtract Rational Expressions 10.4 Solve Rational Equations Unit 10 Review Unit 10 Algebra Skillz 11.1 Parabolas 11.2 Ellipses and Circles 11.3 Hyperbolas 11.4 Classifying Conics Unit 11 Review Unit 11 Algebra Skillz 12.1 Matrix Operations 12.2 Matrix Multiplication 12.3 Inverse Matrices 12.4 Encoding Messages Unit 12 Review SEMESTER 2 EXAM FlippedMath.com Teacher Resources Section 4.1 Solving Systems Graphically File Size: 1138 kb File Type: pdf Practice Solutions File Size: 421 kb File Type: pdf Corrective Assignment File Size: 27 kb File Type: pdf Application Walkthrough ' + ' '; setTimeout(function() { var videoIframe874452072298780253Actual = document.getElementById("video-iframe-874452072298780253").contentDocument; videoIframe874452072298780253Actual.open().write(videoIframe874452072298780253Content); videoIframe874452072298780253Actual.close(); }, 500); FlippedMath.com List of Lessons 0.1 Solving Simple Equations 0.2 Evaluate Equations 0.3 Graph Linear Equations 1.1 Create and Analyze Graphs 1.2 Modeling with Graphs 1.3 Algebraic Properties 1.4 Add and Subtract Polynomials 1.5 Multiply Polynomials Unit 1 Review 2.1 True False Equations 2.2 Solve Equations 2.3 Solve Inequalities 2.4 Rational Equations 2.5 Literal Equations Unit 2 Review 3.1 Standard Form Equations of Lines 3.2 Systems of Inequalities 3.3 Graphing Systems 3.4 Substitution 3.5 Elimination Unit 3 Review 4.1 Describing Data 4.2 Data Distributions 4.3 Boxplots and IQR 4.4 Comparing Distributions Unit 4 Review 5.1 Frequency Tables 5.2 Best Fit Lines 5.3 Correlation Unit 5 Review 6.1 Sequences 6.2 Recursive Formulas 6.3 Explicit Formulas Unit 6 Review 7.1 Exponential Growth 7.2 Exponential Decay 7.3 Linear vs Exponential Unit 7 Review 8.1 Functions, Domain, Range 8.2 Graphs of Functions 8.3 Piecewise Functions 8.4 Graphing Functions to Solve Equations Unit 8 Review 9.1 Greatest Common Factor 9.2 Factor Trinomials 9.3 Factor Trinomials by Grouping 9.4 Multi-Step Factoring Unit 9 Review 10.1 Intro to Quadratics 10.2 Quadratics in Vertex Form 10.3 Quadratics in Standard Form 10.4 Modeling with Quadratics Unit 10 Review 11.1 Simplify Radicals 11.2 Solve Using Square Roots 11.3 Quadratic Formula 11.4 Completing the Square Unit 11 Review Teacher Resources 3.3 Graphing Systems of Equations . File Size: 457 kb File Type: pdf Corrective Assignment File Size: 116 kb File Type: pdf File Size: 118 kb File Type: pdf school Campus Bookshelves menu_book Bookshelves perm_media Learning Objects login Login how_to_reg Request Instructor Account hub Instructor Commons Margin Size Download Page (PDF) Download Full Book (PDF) Periodic Table Physics Constants Scientific Calculator Reference & Cite Tools expand_more Readability selected template will load here This action is not available. 2.4: Graphing Linear Equations- Answers to the Homework Exercises Last updated Save as PDF Page ID 45036 Darlene Diaz Santiago Canyon College via ASCCC Open Educational Resources Initiative \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \) \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\) \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\) \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\) \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vectorC}[1]{\textbf{#1}} \) \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \) \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \) \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \) Graphing and Slope \(\frac{1}{3}\) \(\frac{4}{3}\) \(\frac{1}{2}\) \(-\frac{1}{3}\) \(\frac{16}{7}\) \(-\frac{7}{17}\) \(\frac{1}{16}\) \(\frac{24}{11}\) \(x=\frac{23}{6}\) \(y=-\frac{29}{6}\) Equations of Lines \(y=-\frac{3}{4}x-1\) \(y = −6x + 4\) \(y = − \frac{1}{4} x + 3\) \(y = \frac{1}{3} x + 3\) \(y = −3x + 5\) \(y = − \frac{1}{10} x − \frac{37}{10}\) \(y = \frac{7x}{3} − 8\) \(y = −4x + 3\) \(y = \frac{1}{10} x − \frac{3}{10}\) \(y = − \frac{4}{7} x + 4\) \(y=\frac{5}{2}x\) \(y − (−5) = 9(x − (−1))\) \(y − (−2) = −3(x − 0)\) \(y − (−3) = \frac{1}{5} (x − (−5))\) \(y − 2 = 0(x − 1)\) \(y − (−2) = −2(x − 2)\) \(y − 1 = 4(x − (−1))\) \(y − (−4) = − \frac{2}{3} (x − (−1))\) \(y = − \frac{3}{5} x + 2\) \(y = − \frac{3}{2} x + 4\) \(y = x − 4\) \(y = − \frac{1}{2} x\) \(y = − \frac{2}{3} x − \frac{10}{3}\) \(y = − \frac{5}{2} x − 5\) \(y = −3\) \(y − 3 = −2(x + 4)\) \(y + 2 = \frac{3}{2} (x + 4)\) \(y + 3 = − \frac{8}{7} (x − 3)\) \(y − 5 = − \frac{1}{8} (x + 4)\) \(y + 4 = −(x + 1)\) \(y = − \frac{8}{7} x − \frac{5}{7}\) \(y = −x + 2\) \(y = − \frac{1}{10} x − \frac{3}{2}\) \(y=\frac{1}{3}x+1\) Parallel and Perpendicular Lines \(m_{||} = 2\) \(m_{||} = 1\) \(m_{||} = − \frac{2}{3}\) \(m_{||} = \frac{6}{5}\) \(m_{⊥} = 0\) \(m_{⊥} = −3\) \(m_{⊥} = 2\) \(m_{⊥} = − \frac{1}{3}\) \(y − 4 = \frac{9}{2} (x − 3)\) \(y − 3 = \frac{7}{5} (x − 2)\) \(y + 5 = −(x − 1)\) \(y − 2 = \frac{1}{5} (x − 5)\) \(y − 2 = − \frac{1}{4} (x − 4)\) \(y + 2 = −3(x − 2)\) \(y = −2x + 5\) \(y = − \frac{4}{3} x − 3\) \(y = − \frac{1}{2} x − 3\) \(y = − \frac{1}{2} x − 2\) \(y = x − 1\) \(y=-2x+5\) Unit 1 - Linear Systems Lesson 1 - solving by graphing. Lesson Solutions Worksheet of Practice Problems Worksheet Solutions Lesson 2 - Solving by Substitution Lesson 3 - solving by elimination, lesson 4 - linear system applications, practice test. Copy of Test Use the video to check your solutions Other Unit Materials Lesson Package Review Questions Review Solutions Workbook Solutions If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Algebra basics Unit 1: foundations, unit 2: algebraic expressions, unit 3: linear equations and inequalities, unit 4: graphing lines and slope, unit 5: systems of equations, unit 6: expressions with exponents, unit 7: quadratics and polynomials, unit 8: equations and geometry, review articles. 111 IMAGES Systems Of Linear Equations By Graphing Worksheet Systems Of Linear Equations Graphing Worksheet Graphing Systems of Linear Equations Solving System Of Equations By Graphing Worksheet Graphing Systems Of Linear Equation Worksheet Worksheet Works Graphing Linear Equations 1 B76 VIDEO Lesson 2.1: Graphing Linear Equations (Algebra) McGraw Hill Module6 Lesson 1 Solving Systems of Equations by Graphing Solving Linear Systems by Graphing The geometry of a system of linear equations Algebra 1 Notes 6-1 Systems of Linear Equations: Graphing Lesson 3.1: Solving Systems of Equations by Graphing COMMENTS PDF Unit 5 Unit 5 - Systems of Equations & Inequalities (Updated October 2016) copy. Name: Date: Unit 5: Systems of Equations & Inequalities Homework 1: Solving Systems by Graphing ** This is a 2-page document! ** Solve each system of equations by graphing. Clearly identify your solution. -16 — 6y = 30 9x + = 12 +4 v = —12 O Gina Wilson (All Things ... 5.1: Solve Systems of Equations by Graphing Solutions of a system of equations are the values of the variables that make all the equations true. A solution of a system of two linear equations is represented by an ordered pair (x, y). system of linear equations When two or more linear equations are grouped together, they form a system of linear equations. Quiz 1: Solving Linear Systems by Graphing Flashcards a set of linear equations that share a common point known as the point of intersection (x,y); the solution, (x,y) is an independent and consistent solution. one solution. a system of linear equations that rely on each other for the algebraic or graphic form of the equation. dependent equations. Select the type of equation for the graph described. Unit 5 Unit 5 - Systems of Linear Equations and Inequalities. This unit begins by ensuring that students understand that solutions to equations are points that make the equation true, while solutions to systems make all equations (or inequalities) true. Graphical and substitution methods for solving systems are reviewed before the development of the ... Systems of equations In this unit, we learn how to write systems of equations, solve those systems, and interpret what those solutions mean in a real-world context. If you're seeing this message, it means we're having trouble loading external resources on our website. 01 Get this full course at http://www.MathTutorDVD.comIn this lesson we will learn about systems of equations and how to solve them using the method of graphing... Systems of linear equations Testing a solution to a system of equations. Systems of equations with graphing: y=7/5x-5 & y=3/5x-1. Systems of equations with graphing: exact & approximate solutions. Setting up a system of equations from context example (pet weights) Setting up a system of linear equations example (weight and price) Interpreting points in context of graphs ... Lesson 1 the point that two lines have in common. solution of a linear system. an ordered pair (x,y) that satisfies all the equations in the system. system of linear equations. a set of two or more linear equations that are considered together and involve the same variables. Study with Quizlet and memorize flashcards containing terms like coincide ... 5.1 Solve Systems of Equations by Graphing To solve a system of linear equations by graphing. Step 1. Graph the first equation. Step 2. Graph the second equation on the same rectangular coordinate system. Step 3. Determine whether the lines intersect, are parallel, or are the same line. Step 4. PDF 5.1 Solving Systems of Linear Equations by Graphing Solving a System of Linear Equations by Graphing Step 1 Graph each equation in the same coordinate plane. Step 2 Estimate the point of intersection. Step 3 Check the point from Step 2 by substituting for x and y in each equation of the original system. Solving a System of Linear Equations by Graphing Solve the system of linear equations by ... 4.1.0 Solving Systems of Linear Equations (Graphing) 4.1.0 Solving Systems of Linear Equations (Graphing) Flashcards; Learn; Test; Match; Q-Chat ... Q-Chat; CourtneyWichman Teacher. Top creator on Quizlet. Share. Share. Students also viewed. Coulter World History Unit 1. Teacher 34 terms. mcoulter28. Preview. Scatter Plots: Describing Association ... Solving Systems of Equations - Graphing 1 ... Systems of equations with graphing (video) y=mx+b. In this equation, 'm' is the slope and 'b' is the y-intercept. To graph a line from a slope-intercept equation, take the value of the slope and put it over 1. For example, if the slope was 5, the slope would be 5/1. Next graph the y-intercept, take the number that is the y-intercept, and graph that number on the graph. PDF 6.1 Solving Systems by Graphing Objective: To Solve Systems of Equations by Graphing Content Standard: A.REI.6 Section 6.1 4 System of Equations - Two or more linear equations form a system of equations. - Any ordered pair that makes all of the equations in a system true is a solution of a system of linear equations. 4.1: Solving Systems by Graphing Solve the following system of equations: \[3x-5y =-15 \\ 2x+y =-4 \label{system2}\] Solution. Once again, we are looking for the point that satisfies both equations of the System \ref{system2}. Thus, we need to find the point that lies on the graphs of both lines represented by the equations of System \ref{system2}. Illustrative Mathematics Algebra 1, Unit 2.12 Preparation Students write a system of equations to represent the quantities and constraints in each of several situations, find a solution that meets multiple constraints by graphing, and then interpret the solution in context. In the process, they reason both abstractly and quantitatively (MP2). As they analyze relationships mathematically and reflect on ... 4.1 Solving Systems by Graphing A2.5.4 Solve systems of linear equations and inequalities in two variables by substitution, graphing, and use matrices with three variables; Section 4.1 Solving Systems Graphically. ... 1138 kb: File Type: pdf: Download File. Practice Solutions. a2_4.1_solutions.pdf: File Size: 421 kb: File Type: pdf: Download File. Corrective Assignment. a2_4 ... 3.3 Graphing Systems Common Core Standard: A-REI.C.6, A-REI.D.10, and A-REI.D.12. Student Outcome: Students will solve systems of linear equations graphically, understanding there are three different types of solutions. Students will also use technology (graphing calculator) to solve a system by graphing. Algebra 1: Unit 5 Systems of Linear Equations & Inequalities A set of systems that have different slopes. Dependent. A set of systems that have the same slope. Solution (s) The ordered pair (s) that is true for all of the equations. Intersection. A point where lines intersect. Also the solution of a consistent independent system. Elimination Method. 2.4: Graphing Linear Equations- Answers to the Homework Exercises y = −2x + 5 y = − 2 x + 5. This page titled 2.4: Graphing Linear Equations- Answers to the Homework Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Darlene Diaz ( ASCCC Open Educational Resources Initiative) via source content that was edited to the style and standards of the LibreTexts ... Desmos Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. Graphing Systems of Equations Flashcards the slopes of equations are 2. the system has no solution. The solution of a system. a point where the lines intersect. the solution for the system is (2, 5) Graph to find a solution y = -x + 7. y = 2x + 1. the solutions is (6, 3) graph to find a solution y = 2/3 x - 1. Linear equations and linear systems Linear equations and linear systems: Unit test; Lesson 3: Balanced moves. Learn. Intro to equations with variables on both sides (Opens a modal) Equations with variables on both sides: 20-7x=6x-6 ... Systems of equations with graphing: y=7/5x-5 & y=3/5x-1 (Opens a modal) Practice. Grade 10 Math Unit 1 Free lessons, worksheets, and video tutorials for students and teachers. Topics in this unit include: solving linear systems by graphing, substitution, elimination, and solving application questions. This follows chapter 1 of the principles of math grade 10 McGraw Hill textbook. Solving equations & inequalities Unit 1. Algebra foundations. Unit 2. Solving equations & inequalities. Unit 3. Working with units ... Linear equations & graphs. Unit 5. Forms of linear equations. Unit 6. Systems of equations. Unit 7. Inequalities (systems & graphs) Unit 8. Functions. Unit 9. Sequences. Unit 10. Absolute value & piecewise functions. ... Reasoning with linear ... Algebra Basics Linear equations and inequalities: Unit test; Unit 4 Unit 4: Graphing lines and slope. Quadrants on the coordinate plane; Solutions to 2-variable equations; ... Systems of equations with graphing; Systems of equations with elimination; Systems of equations with elimination challenge; assignment essay homework paper phd report review speech thesis